In an experiment in which helium nuclei are sent against a target consisting of \(1 \mathrm{~mm}\) of gold, it is observed that \(99.9 \%\) reaches a detector that is on the beam line. Assuming that all the atomic mass is concentrated in the spherical-shaped nucleus, and neglecting the size of the helium nucleus, determine the radius of the gold nucleus. Such an experiment was used by Rutherford to determine the size of nuclei. Make use of the dimensional analysis and the number \(n\) determined in the question 3 .

Question 3 Determine the number \(n\) of atoms per \(\mathrm{cm}^{3}\) present in gold \((Z=79, A=197\), density 19.3 \(\mathrm{g} / \mathrm{cm}^{3}\) ) and estimate the distance between the centers of two atoms, assuming the atoms are arranged on a cubic lattice.